Chapter 7 - Eigenvalues and Eigenvectors - 7.3 Symmetric Matrices and Orthogonal Diagonalization - 7.3 Exercises - Page 370: 3

not symmetric.

A matrix is symmetric if it is equal to its transpose. Here the transpose of the matrix is $\begin{bmatrix} 4& 3& 1 \\ -2& 1&2\\ 1&2&1\\ \end{bmatrix} $, which is not the same as the original matrix; thus it is not symmetric.